In the xy -plane, the graph of the equation y = - x 2 + 9 x - 100 intersects

Question

In the xy -plane, the graph of the equation y = - x 2 + 9 x - 100 intersects the line y = c at exactly one point. What is the value of c ?

Accepted Answer

Choice C is correct. In the xy -plane, the graph of the line y = c is a horizontal line that crosses the y -axis at y = c and the graph of the quadratic equation y = - x 2 + 9 x - 100 is a parabola. A parabola can intersect a horizontal line at exactly one point only at its vertex. Therefore, the value of c should be equal to the y -coordinate of the vertex of the graph of the given equation. For a quadratic equation in vertex form, y = a x - h 2 + k , the vertex of its graph in the xy -plane is h , k . The given quadratic equation, y = - x 2 + 9 x - 100 , can be rewritten as y = - x 2 - 2 9 2 x + 9 2 2 + 9 2 2 - 100 , or y = - x - 9 2 2 + - 319 4 . Thus, the value of c is equal to - 319 4 . Choice A is incorrect and may result from conceptual or calculation errors. Choice B is incorrect and may result from conceptual or calculation errors. Choice D is incorrect and may result from conceptual or calculation errors.

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